ON k-STRONG DISTANCE IN STRONG DIGRAPHS

For a nonempty set S of vertices in a strong digraph D, the strong distance d(S) is the minimum size of a strong subdigraph of D containing the vertices of S. If S contains k vertices, then d(S) is referred to as the k-strong distance of S. For an integer k > 2 and a vertex v of a strong digraph D, the k-strong eccentricity sek(v) of v is the maximum k-strong distance d(S) among all sets S of k vertices in D containing v. The minimum k-strong eccentricity among the vertices of D is its k-strong radius sradk D and the maximum k-strong eccentricity is its k-strong diameter sdiamk D. The k-strong center (k-strong periphery) of D is the subdigraph of D induced by those vertices of k-strong eccentricity sradk(D) (sdiamk(D)). It is shown that, for each integer k > 2, every oriented graph is the k-strong center of some strong oriented graph. A strong oriented graph D is called strongly k-self-centered if D is its own k-strong center. For every integer r > 6, there exist infinitely many strongly 3-self-centered oriented graphs of 3-strong radius r. The problem of determining those oriented graphs that are k-strong peripheries of strong oriented graphs is studied.